User blog:Rgetar/Ordinal subtraction and integer extension of ordinals
Note: on November 8, 2017 this blog was edited: were added sections "wet "integer ordinals"", "Left, right and central successors and predecessors", "List of not wet "integer ordinals"", and was modified section"List of wet "integer ordinals"". Ordinal subtraction is operation opposite to ordinal addition. Ordinal addition is non-commutative, so, there are two ordinal subtractions (as for exponentiation, which is also non-commutative and also has two opposite operations: root and logarithm): 1. α-β: (α-β)+β = α 2. -β+α: β+(-β+α) = α (Note: in Ordinals array function blog I denoted both subtractions as α-β, and later I started to denote second subtraction as -β+α). As we can extend natural numbers and zero to integer numbers using subtraction, we can extend ordinals to "integer ordinals". Opposite ordinals Subtraction of β we may describe as addition of opposite "integer ordinal" -β: α-β = α+(-β) -β+α = (-β)+α -α is opposite to α: α+(-α) = 0 (-α)+α=0 To get opposite to α we need write Cantor normal form of α backwards and change all its coefficients to opposite: α = ωα0·n0 + ωα1·n1 + ωα2·n2 + ... -α = ... + ωα2·(-n2) + ωα1·(-n1) + ωα0·(-n0) Examples: α = ω+1 -α = -1-ω α = ω8·4+ω3·2+1 -α = -1-ω3·2-ω8·4 Addition rules Extension of addition rules on "integer ordinals" (see also Ordinal arithmetic blog, How to add two ordinals section): 1. ωα1·n1+ωα2·n2 = ωα1·(n1+n2) (if α1 = α2) 2. ωα1·n1+ωα2·n2 = ωα1·n1+ωα2·n2 (if (α1 > α2 and n2 > 0) or (α1 < α2 and n2 < 0)) 3. ωα1·n1+ωα2·n2 = ωα2·n2 (if α1 < α2 and n2 > 0) 4. ωα1·n1+ωα2·n2 = ωα1·n1 (if α1 > α2 and n1 < 0) So, ωαi·ni "eats" all terms leftward until it faces a term with larger αi, if ni > 0, and rightward, if ni < 0. "Mixed integer ordinals" There are "fully positive integer ordinals" with all coefficients of Cantor normal form positive (they are ordinals), such as ω+1 And there are "fully negative integer ordinals" with all coefficients of Cantor normal form negative (they are opposite to ordinals), such as -1-ω There are also "mixed integer ordinals" with some coefficients of Cantor normal form positive, and some negative, such as ω-1 1-ω They are, however, also positive (> 0) or negative (< 0), but not "fully": ω-1 is positive, but not fully positive 1-ω is negative, but not fully negative fpp, fnp An "integer ordinal" α can be uniquely represented as difference of two ordinals: fully positive part (fpp) of α and fully negative part (fnp) of α. α = fpp(α)-fnp(α) Example: α = ω8 + ω5 + ω4 + ω3 + ω - 1 - ω - ω2 - ω6 - ω7 fpp(α) = ω8 + ω5 + ω4 + ω3 + ω fnp(α) = ω7 + ω6 + ω2 + ω + 1 Properties: if fpp(α) = 0, fnp(α) = 0 then α = 0 (it is also an ordinal) if fpp(α) > 0, fnp(α) = 0 then α is a "fully positive ordinal" (or just an ordinal) if fpp(α) = 0, fnp(α) > 0 then α is a "fully negative ordinal" if fpp(α) > 0, fnp(α) > 0 then α is a "mixed ordinal" if fpp(α) > fnp(α) then α is positive if fnp(α) > fpp(α) then α is negative -α = fnp(α)-fpp(α) wet "integer ordinals" wet means "without equal terms" wet "integer ordinal" is "integer ordinal" such as there is no term of Cantor normal form of its fpp equal to term of Cantor normal form of its fnp except, maybe, finite terms of Cantor normal forms of fpp and fnp (considering that Cantor normal form can be represented without coefficients). Examples: ω22 - 1 - ω is wet since its fpp ω22 and its fnp ω + 1 have no equal terms. ω22 - 1 - ω2 is not wet since its fpp ω22 and its fnp ω2 + 1 have equal term ω2. So, in a not wet integer ordinal there is a term which presents at least twice (with different coefficients), and in a wet integer ordinal all terms may present once. A finite term always may present once. Even if there are finite terms in fpp and fnp, they are subtracted. Left, right and central successors and predecessors Right successor of integer ordinal α is α + 1 For example, 0 is right successor of -1 ω is right successor of ω - 1 Right predecessor is α - 1 Left successor is 1 + α For example, -ω is left successor of -1 - ω Left predecessor is -1 + α Central successor is α with its finite part increased by 1. Central predecessor is α with its finite part decreased by 1. α always has one right successor, one right predecessor, one left successor, one left predecessor. If α has not finite part, then α has infinitely many central successors and infinitely many central predecessors. For example, 0 has infinitely many central successors: 1 ω + 1 - ω ω2 + 1 - ω2 ω2 + 1 - ω2 ω2 + ω + 1 - ω - ω2 etc. ω also has infinitely many central successors: ω + 1 ω2 + 1 - ω ω3 + 1 - ω2 etc. but ω + 1 - ω has only one central successor ω + 2 - ω If α has finite part, then α has one central successor and one central predecessor. If fnp(α) is finite, right successor and predecessor of α are not equal to α and equal to central successor and predecessor of α respectively. If fnp(α) is infinite, right successor and predecessor of α are equal to α and not equal to central successor and predecessor of α respectively. If fpp(α) is finite, left successor and predecessor of α are not equal to α and equal to central successor and predecessor of α respectively. If fpp(α) is infinite, left successor and predecessor of α are equal to α and not equal to central successor and predecessor of α respectively. List of wet "integer ordinals" wet "integer ordinals" may be positioned in ascending order. List of all wet "integer ordinals" with Cantor normal form coefficients -1 or 1 and degrees of ω less than 3 in ascending order: -1-ω-ω2 -ω-ω2 1-ω-ω2 -1-ω2 -ω2 1-ω2 ω-1-ω2 ω-ω2 ω+1-ω2 -1-ω -ω 1-ω -1 0 1 ω-1 ω ω+1 ω2-1-ω ω2-ω ω2+1-ω ω2-1 ω2 ω2+1 ω2+ω-1 ω2+ω ω2+ω+1 List of not wet "integer ordinals" List of all not wet "integer ordinals" with Cantor normal form coefficients -1 or 1 and degrees of ω less than 3: ω-1-ω-ω2 ω+1-ω-ω2 ω-1-ω ω+1-ω ω2-1-ω2 ω2+1-ω2 ω2+ω-1-ω-ω2 ω2+ω+1-ω-ω2 ω2+ω-1-ω ω2+ω+1-ω Category:Blog posts